Atlas › Modular forms › Weight 2 › Level 730 › 730.2.a

Newspace 730.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(730))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
730.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} + q^{4} + q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	730g1 rank 1
730.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$	730b1 rank 0
730.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$	730d1 rank 0
730.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} - 2q^{7} + O(q^{8})$	+ + −	$+1$	730a1 rank 0
730.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + 2q^{7} + O(q^{8})$	+ − −	$-1$	730f1 rank 1
730.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} + q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$	730e1 rank 0
730.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + 3q^{3} + q^{4} - q^{5} - 3q^{6} + q^{7} + O(q^{8})$	+ + −	$+1$	730c1 rank 0
730.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	730h1 rank 0
730.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	730i1 rank 1
730.2.a.j	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	730j1 rank 1
730.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$	730k1 rank 0
730.2.a.l	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - q^{3} + 2q^{4} + 2q^{5} + q^{6} - 8q^{7} + O(q^{8})$	+ − −	$-1$
730.2.a.m	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} + q^{7} + O(q^{8})$	+ + +	$-1$
730.2.a.n	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	− + +	$+1$
730.2.a.o	4	degree 4	$4q + 4q^{2} - q^{3} + 4q^{4} + 4q^{5} - q^{6} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
730.2.a.a	$x - 19$
730.2.a.b	$x + 69$
730.2.a.c	$x - 77$
730.2.a.d	$x - 36$
730.2.a.e	$x + 92$
730.2.a.f	$x - 29$
730.2.a.g	$x - 21$
730.2.a.h	$x - 45$
730.2.a.i	$x + 41$
730.2.a.j	$x + 121$
730.2.a.k	$x + 51$
730.2.a.l	$x^{2} + 60x + 68$
730.2.a.m	$x^{3} + 43x^{2} - 1810x - 77756$
730.2.a.n	$x^{3} - 143x^{2} + 6186x - 72556$
730.2.a.o	$x^{4} - 119x^{3} + 2644x^{2} + 51888x - 1530176$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(730))$	23
Old forms	84
Cusp forms $S_{2}(\Gamma_0(730))$	107
Eisenstein series $E_{2}(\Gamma_0(730))$	7
Modular forms $M_{2}(\Gamma_0(730))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(730)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(146))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(365))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.795 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 5, 10, 73, 146, 365, 730]: [0, 1, 1, 3, 6, 20, 38, 114] (certified at ell = 2147482873)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 6 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(730, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 730:2 1000

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